94 V NORMED SPACES

(/„) is a Cauchy sequence which does not converge (if there existed a limit g of (/„) in E,
show that one would necessarily have g(t) = 1 for 0 ^ / ^ i and #(0 = 0 for J < t ^ 1,
which would violate the continuity of g).

(b) Show that the distance on E defined in (5.1.4) is not topologically equivalent to the
distance defined in (3.2.4). (Give an example of a sequence in E which tends to 0 for
\\f-ff\\i, but has no limit for the distance defined in (3.2.4).)

2. If A, B are two subsets of a normed space E, we denote by A -f- B the set of all sums
a 4- 6, where a e A, b e B.

(a) Show that if one of the sets A, B is open, A + B is open.

(b) Show that if both A and B are compact, A + B is compact (use (3.17.9) and
(3.20.16)).

(c) Show that if A is compact and B is closed, then A 4- B is closed.

(d) Give an example of two closed subsets A, B of R such that A + B is not closed
(cf. the example given before (3.4.1)).

3. Let E be a normed space.

(a) Show that in E the closure of an open ball is the closed ball of same center and
same radius, the interior of a closed ball is the open ball of same center and same
radius, and the frontier of an open ball (or of a closed ball) is the sphere of same center
and same radius (compare to Section 3.8, Problem 4).

(b) Show that the open ball B(0; r) is homeomorphic to E (consider the mapping
x-*rxl(l+\\x\\)).

4. In a normed space E, a segment is the image of the interval [0,1 ] of R by the continuous
mapping / -> ta + (1 — t)b, where a e E and b e E; a and b are called the extremities of
the segment. A segment is compact and connected. A broken line in E is a subset L of E
such that there exists a finite sequence (jtf)o^ iĞĞ of points of E having the property that
if Sf is the segment of extremities xt and xt+1 for 0 *£ / ^ n — 1, L is the union of the S4;
the sequence (xt) is said to define the broken line L (a given broken line may be defined
in general by infinitely many finite sequences). If A is a subset of E, a, b two points of A,
one says that a and b are linked by a broken line in A, if there is a sequence (xi)o*i*n
such that a = x0, b = xn and that the broken line L defined by that sequence is con-
tained in A.

If any two points of A can be linked by a broken line in A, A is connected. Conversely,
if A c E is a connected open set, show that any two points of A can be linked by a
broken line in A (prove that the set of points y e A which can be linked to a given point
a e A by a broken line in A is both open and closed in A).

5. In a real vector space E, a linear variety V is a set of the form a + M, where M is a
linear subspace of E; the dimension (resp. codimension) of V is by definition the dimen-
sion (resp. codimension) of M. If b $ V and if V has finite dimension p (resp. finite
codimension q)9 the smallest linear variety W containing both b and V has finite dimen-
sion^ + 1 (resp. finite codimension g — 1).

Let A be an open connected subset of a real normed space E, and let (Vn) be a denumer-
able sequence of linear varieties in E, each of which has codimension ^ 2; show that
if B is the union of the Vfl, A n (E — B) is connected. (Hint: Use Problem 4; if L
is a broken line linking two points a, b of A n (E — B) in A, prove that there exists an-
other broken line L' "close" to L, contained in A n (E — B). To do that, observe that
if x e E — B, the set of points y e E such that the segment of extremities x, y does not
meet any Vn, is dense in E, using (2.2.17).)

In particular, if the dimension of E is 5*2, and if D is a denumerable subset of E,
A n (E — D) is connected.